Motivated by a Möbius invariant subdivision scheme for polygons, we study a curvature notion for discrete curves where the cross-ratio plays an important role in all our key definitions. Using a particular Möbius invariant point-insertion-rule, comparable to the classical four-point-scheme, we construct circles along discrete curves. Asymptotic analysis shows that these circles defined on a sampled

The Olver–Benney equation is a nonlinear fifth-order equation, which describes the interaction effects between short and long waves. In this paper, we prove the global existence of solutions of the Cauchy problem associated with this equation.

We prove that the Euclidean ball can be realized as a Fatou component of a holomorphic automorphism of \(\mathbb{C}^m\), in particular as the escaping and the oscillating wandering domain. Moreover, the same is true for a large class of bounded domains, namely for all bounded simply connected regular open sets \(\Omega \subset \mathbb{C}^m\) whose closure is polynomially convex. Our result gives in

We consider the complex hyperbolic quadric \({Q^*}^n\) as a complex hypersurface of complex anti-de Sitter space. Shape operators of this submanifold give rise to a family of local almost product structures on \({Q^*}^n\), which are then used to define local angle functions on any Lagrangian submanifold of \({Q^*}^n\). We prove that a Lagrangian immersion into \({Q^*}^n\) can be seen as the Gauss map

Given any diagonal cyclic subgroup \(\Lambda \subset \text {GL}(n+1,k)\) of order d, let \(I_d\subset k[x_0,\ldots , x_n]\) be the ideal generated by all monomials \(\{m_{1},\ldots , m_{r}\}\) of degree d which are invariants of \(\Lambda\). \(I_d\) is a monomial Togliatti system, provided \(r \le \left( {\begin{array}{c}d+n-1\\ n-1\end{array}}\right)\), and in this case the projective toric variety

In this note, we formulate an explicit formula for the completed zeta function of a function field K of genus g over a finite field \(\mathbb {F}_q\), analogous to the Weil explicit formula. Then we give an arithmetic formula for the nth Li coefficient \(\lambda _{K}(n)\) for the function field K. Furthermore, as an application, we give some formulas for the Euler–Stieltjes constants \(\gamma _{K}\)

We study the functional calculus associated with a hypoelliptic left-invariant differential operator \(\mathcal {L}\) on a connected and simply connected nilpotent Lie group G with the aid of the corresponding Rockland operator \(\mathcal {L}_0\) on the ‘local’ contraction \(G_0\) of G, as well as of the corresponding Rockland operator \(\mathcal {L}_\infty \) on the ‘global’ contraction \(G_\infty

We discuss new sufficient conditions under which an affine manifold \((M,\nabla )\) is geodesically connected. These conditions are shown to be essentially weaker than those discussed in groundbreaking work by Beem and Parker and in recent work by Alexander and Karr, with the added advantage that they yield an elementary proof of the main result.

We consider the compressible Euler system describing the motion of an ideal fluid confined to a straight layer \(\Omega _{\delta }=(0,\delta )\times {\mathbb {R}}^2, \ \ \delta >0\). In the framework of dissipative measure-valued solutions, we show the convergence to the strong solution of the 2D incompressible Euler system when the Mach number tends to zero and \(\delta \rightarrow 0\).

In this paper, we focus on the standing waves with prescribed mass for the coupled Hartree–Fock system, which is the basic quantum chemistry model of small number of electrons interacting with static nuclei. This leads to study the normalized solutions to the following nonlocal elliptic system $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u+(x_{1}^{2}+x_{2}^{2}+\cdot \cdot \cdot

The aim of this paper is to obtain a generalized CK-extension theorem in superspace for the biaxial Dirac operator \(\partial _{\mathbf{x}} +\partial _{\mathbf{y}}\). In the classical commuting case, this result can be written as a power series of Bessel type of certain differential operators acting on a single initial function. In the superspace setting, novel structures appear in the cases of negative

We study local (the heat equation) and nonlocal (convolution-type problems with an integrable kernel) evolution problems on a metric connected finite graph in which some of the edges have infinity length. We show that the asymptotic behaviour of the solutions to both local and nonlocal problems is given by the solution of the heat equation, but on a star-shaped graph in which there are only one node

We show that the Hodge numbers of Sasakian manifolds are invariant under arbitrary deformations of the Sasakian structure. We also present an upper semi-continuity theorem for the dimensions of kernels of a smooth family of transversely elliptic operators on manifolds with homologically orientable transversely Riemannian foliations. We use this to prove that the \(\partial {\bar{\partial }}\)-lemma

We characterize global Denjoy–Carleman hypoellipticity for the family of systems acting on the torus \({\mathbb {T}}_{t}^{N} \times {\mathbb {T}}_{x}\) given by \(L_{j} = \frac{\partial }{\partial t_{j}} + \left( a_{j}(t_{j}) + ib_{j}(t_{j}) \right) \frac{\partial }{\partial x} + \lambda _j\) , where \(a_j, b_j\) are real-valued \(2\pi\)-periodic elements of the classes and \(j = 1, 2,\ldots , N\)

We study magnetic trajectories in the unit tangent sphere bundle with pseudo-Riemannian g-natural metrics of a Riemannian manifold. A high interest is dedicated to the case when the base manifold is a space form and when the metric is of Kaluza–Klein type. Slant curves are obtained when a certain conservation law is satisfied. We give a complete classification of slant magnetic curves (respectively

We show that every finite group of order divisible by 2 or q, where q is a prime number, admits a \(\{2, q\}'\)-degree nontrivial irreducible character with values in \({\mathbb{Q}}(e^{2 \pi i /q})\). We further characterize when such character can be chosen with only rational values in solvable groups. These results follow from more general considerations on groups admitting a \(\{p, q\}'\)-degree

We provide geometric estimates for conformal maps of the unit disc. As a consequence, a bound for the derivative of such maps follows yielding, in particular, an answer to a question posed by Martín Chuaqui. Finally, estimates regarding the curvature of image arcs under conformal maps are also obtained.

In this paper, we establish various maximum principles and develop the method of moving planes for equations involving the uniformly elliptic nonlocal Bellman operator. As a consequence, we derive multiple applications of these maximum principles and the moving planes method. For instance, we prove symmetry, monotonicity and uniqueness results and asymptotic properties for solutions to various equations

In this paper, we consider the semilinear Cauchy problem for the heat equation with power nonlinearity in the Heisenberg group \(\mathbf {H}_n\). The heat operator is given in this case by \(\partial _t-\varDelta _{{{\,\mathrm{H}\,}}}\), where \(\varDelta _{{{\,\mathrm{H}\,}}}\) is the so-called sub-Laplacian on \(\mathbf {H}_n\). We prove that the Fujita exponent \(1+2/Q\) is critical, where \(Q=2n+2\)

In this paper, we introduce operators that are represented by upper triangular \(2\times 2\) block matrices whose entries satisfy some algebraic constraints. We call them Brownian-type operators of class \({\mathcal {Q}},\) briefly operators of class \({\mathcal {Q}}.\) These operators emerged from the study of Brownian isometries performed by Agler and Stankus via detailed analysis of the time shift

In this paper, we are concerned with the following question: if the tensor product of finitely generated modules M and N over a local complete intersection domain is maximal Cohen–Macaulay, then must M or N be a maximal Cohen–Macaulay? Celebrated results of Auslander, Lichtenbaum, and Huneke and Wiegand yield affirmative answers to the question when the ring considered has codimension zero or one,

A coprime commutator in a profinite group G is an element of the form [x, y], where x and y have coprime order and an anti-coprime commutator is a commutator [x, y] such that the orders of x and y are divisible by the same primes. In the present paper, we establish that a profinite group G is finite-by-pronilpotent if the cardinality of the set of coprime commutators in G is less than \(2^{\aleph _0}\)

In this article, we characterize all eigenfunctions of the Laplacian on homogeneous trees, which are the Poisson transform of \(L^p\) functions defined on the boundary. Using the duality argument, we also proved the restriction theorem for the Helgason–Fourier transforms on a homogeneous tree.

Symmetry plays a basic role in variational problems (settled, e.g., in \({\mathbb {R}}^{n}\) or in a more general manifold), for example, to deal with the lack of compactness which naturally appears when the problem is invariant under the action of a noncompact group. In \({\mathbb {R}}^n\), a compactness result for invariant functions with respect to a subgroup G of \(\mathrm {O}(n)\) has been proved

In this paper, we shall be concerned with evaluation of Hewitt–Stromberg measure of Cartesian product sets by means of the measure of their components. This is done by the construction of new multifractal measures, on the Euclidean space \({\mathbb {R}}^n\), in a similar manner to Hewitt–Stromberg measures but using the class of all n-dimensional half-open binary cubes of covering sets in the definition

We prove an existence result for the Poisson equation on non-compact Riemannian manifolds satisfying weighted Poincaré inequalities outside compact sets. Our result applies to a large class of manifolds including, for instance, all non-parabolic manifolds with minimal positive Green’s function vanishing at infinity. On the source function, we assume a sharp pointwise decay depending on the weight appearing

We study the Hermitian curvature flow of locally homogeneous non-Kähler metrics on compact complex surfaces. In particular, we characterize the long-time behavior of the solutions to the flow. We also provide the first example of a compact complex non-Kähler manifold admitting a finite time singularity for the Hermitian curvature flow. Finally, we compute the Gromov–Hausdorff limit of immortal solutions

We are interested in differential forms on mixed-dimensional geometries, in the sense of a domain containing sets of d-dimensional manifolds, structured hierarchically so that each d-dimensional manifold is contained in the boundary of one or more \(d + 1\)-dimensional manifolds. On any given d-dimensional manifold, we then consider differential operators tangent to the manifold as well as discrete

We give a notion of “combinatorial proximity” among strongly stable ideals in a given polynomial ring with a fixed Hilbert polynomial. We show that this notion guarantees “geometric proximity” of the corresponding points in the Hilbert scheme. We define a graph whose vertices correspond to strongly stable ideals and whose edges correspond to pairs of adjacent ideals. Every term order induces an orientation

We study the space of derivations for some finite-dimensional evolution algebras, depending on the twin partition of an associated directed graph. For evolution algebras with a twin-free associated graph, we prove that the space of derivations is zero. For the remaining families of evolution algebras, we obtain sufficient conditions under which the study of such a space can be simplified. We accomplish

A remarkable result of Thompson states that a finite group is soluble if and only if all its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory of groups, aiming for global properties of groups from local properties of two-generated (or more generally, n-generated) subgroups. We contribute an extension

We analyze some properties of a class of multiexponential maps appearing naturally in the geometric analysis of Carnot groups. We will see that such maps can be useful in at least two interesting problems: first, in relation to the analysis of some regularity properties of horizontally convex sets. Then, we will show that our multiexponential maps can be used to prove the Pansu differentiability of

Let V be a vector bundle over a smooth curve C. In this paper, we study twisted Brill–Noether loci parametrising stable bundles E of rank n and degree e with the property that \(h^0 (C, V \otimes E) \ge k\). We prove that, under conditions similar to those of Teixidor i Bigas and of Mercat, the Brill–Noether loci are nonempty and in many cases have a component which is generically smooth and of the

In this paper we develop fundamental theories for a scalar first-order hyperbolic partial differential equation with two internal variables which models single-species population dynamics with two physiological structures such as age–age, age–maturation, age–size, and age–stage. Classical techniques of treating structured models with a single internal variable are generalized to study the double physiologically

Consideration in this paper is three-dimensional capillary–gravity water flows governed by the geophysical water wave equations with all the Coriolis terms being retained. It is proved that the merely possible flow exhibiting a constant vorticity vector captures vanishing vertical velocity, constant horizontal velocity and flat free surface.

We consider the scaling of the optimal constant in Korn’s first inequality for elliptic and parabolic shells which was first given by Grabovsky and Harutyunyan with hints coming from the test functions constructed by Tovstik and Smirnov on the level of formal asymptotic expansions. Here, we employ the Bochner technique in Remannian geometry to remove the assumption that the middle surface of the shell

We prove ultradifferentiable Chevelley restriction theorems for a wide range of ultradifferentiable classes. As a special case we find that isotropic functions, i.e., functions defined on the vector space of real symmetric matrices invariant under the action of the special orthogonal group by conjugation, possess some ultradifferentiable regularity if and only if their restriction to diagonal matrices

We show the existence of non-Einstein homogeneous critical metrics for any quadratic curvature functional in dimension three.

A C0-Finsler structure is a continuous function \(F:TM \rightarrow [0,\infty )\) defined on the tangent bundle of a differentiable manifold M such that its restriction to each tangent space is an asymmetric norm. We use the convolution of F with the standard mollifier in order to construct a mollifier smoothing of F, which is a one-parameter family of Finsler structures \(F_\varepsilon\) that converges

We prove the existence of nonnegative variational solutions to the obstacle problem associated with the degenerate doubly nonlinear equation $$\begin{aligned} \partial _t b(u) - {{\,\mathrm{div}\,}}(Df(Du)) = 0, \end{aligned}$$ where the nonlinearity \(b :\mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}\) is increasing, piecewise \(C^1\) and satisfies a polynomial growth condition. The prototype

We consider the semi-classical limit of the quantum evolution of Gaussian coherent states whenever the Hamiltonian H is given, as sum of quadratic forms, by \( H= -\frac{{\hbar ^{2}}}{2m}\,\frac{d^{2}\,}{dx^{2}}\,\dot{+}\,\alpha \delta _{0}\), with \(\alpha \in \mathbb R\) and \(\delta _{0}\) the Dirac delta-distribution at \(x=0\). We show that the quantum evolution can be approximated, uniformly

For a complete symplectic manifold \(M^{2n}\), we define the \(L^{2}\)-hard Lefschetz property on \(M^{2n}\). We also prove that the complete symplectic manifold \(M^{2n}\) satisfies \(L^{2}\)-hard Lefschetz property if and only if every class of \(L^{2}\)-harmonic forms contains a \(L^{2}\) symplectic harmonic form. As an application, we get if \(M^{2n}\) is a closed symplectic parabolic manifold

It has been known for some time that there exist 5 essentially different real forms of the complex affine Kac–Moody algebra of type \(A_2^{(2)}\) and that one can associate 4 of these real forms with certain classes of “integrable surfaces,” such as minimal Lagrangian surfaces in \(\mathbb {CP}^2\) and \(\mathbb {CH}^2\), as well as definite and indefinite affine spheres in \({\mathbb {R}}^3\). In

We study existence and uniqueness of nonnegative solutions to a problem which is modeled by $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta _p u = u^{-\theta }|\nabla u|^p + fu^{-\gamma }& \text {in}\, \Omega , \\ u=0 & \text {on}\ \partial \Omega , \end{array}\right. } \end{aligned}$$ where \(\Omega\) is an open bounded subset of \({\mathbb {R}}^N\) (\(N\ge 2\)), \(\Delta _p\) is

In this work, we establish the existence of nonzero solutions for a class of quasilinear elliptic equations involving indefinite nonlinearities with exponential critical growth of Trudinger–Moser type. Our proofs rely on variational arguments in a Orlicz–Sobolev space with a version of the Trudinger–Moser inequality.

In this paper, we study balanced metrics and Berezin quantization on a class of Hartogs domains defined by \(\varOmega _n=\{(z_1,\ldots ,z_n)\in {\mathbb {C}}^n:\vert z_1\vert<\vert z_2\vert<\cdots<\vert z_n\vert <1\}\) which generalize the so-called classical Hartogs triangle. We introduce a Kähler metric \(g(\nu )\) associated with the Kähler potential \(\varPhi _n(z):=-\sum _{k=1}^{n-1}\nu _k\ln

Motivated by the Hilbert-space model for quantum mechanics, we define a pre-Hilbert space logic to be a pair \((S,{\mathscr {L}})\), where S is a pre-Hilbert space and \({\mathscr {L}}\) is an orthocomplemented poset of orthogonally closed linear subspaces of S, closed w.r.t. finite-dimensional perturbations (i.e., if \(M\in {\mathscr {L}}\) and F is a finite-dimensional linear subspace of S, then

Fueter’s theorem (1934) asserts that every holomorphic intrinsic function of one complex variable induces an axial quaternionic monogenic function. Sce (Atti Accad Naz Lincei Rend Cl Sci Fis Mat Nat 23:220–225, 1957) generalizes Fueter’s theorem to the Euclidean spaces \({{\mathbb {R}}}^{n+1}\) for n being odd positive integers. By using pointwise differential computation he asserted that every holomorphic

The X-rank of a point p in projective space is the minimal number of points of an algebraic variety X whose linear span contains p. This notion is naturally submultiplicative under tensor product. We study geometric conditions that guarantee strict submultiplicativity. We prove that in the case of points of rank two, strict submultiplicativity is entirely characterized in terms of the trisecant lines

We construct an unbounded strictly pseudoconvex Kobayashi hyperbolic and complete domain in \({\mathbb {C}}^2\), which also possesses complete Bergman metric, but has no nonconstant bounded holomorphic functions.

This paper is a contribution to frame theory. Frames in a Hilbert space are generalizations of orthonormal bases. In particular, Gabor frames of \(L^2(\mathbb {R})\), which are made of translations and modulations of one or more windows, are often used in applications. More precisely, the paper deals with a question posed in the last years by Christensen and Hasannasab about the existence of overcomplete

This paper establishes a connection between a problem in Potential Theory and Mathematical Physics, arranging points so as to minimize an energy functional, and a problem in Combinatorics and Number Theory, constructing ’well-distributed’ sequences of points on [0, 1). Let \(f:[0,1] \rightarrow {\mathbb {R}}\) be (1) symmetric \(f(x) = f(1-x)\), (2) twice differentiable on (0, 1), and (3) such that

We assume to have information about the generating properties of the subsets of a finite group G. In particular, we consider the two following situations. We know, for every subset X of G, whether X is a generating set of G. We know the graph whose vertices are the subsets of G and in which there is an edge connecting X and Y if and only if \(X\cup Y\) is a generating set of G. We discuss how this

We extended the study of the linear fractional self maps (e.g., by Cowen–MacCluer and Bisi–Bracci on the unit balls) to a much more general class of domains, called generalized type I domains, which includes in particular the classical bounded symmetric domains of type I and the generalized balls. Since the linear fractional maps on the unit balls are simply the restrictions of the linear maps of the

We prove that if the elliptic problem \(-\Delta u+b(x)|\nabla u|=c(x)u\) with \(c\ge 0\) has a positive supersolution in a domain \(\varOmega \) of \( {\mathrm {R}}^{N\ge 3}\), then c, b must satisfy the inequality $$\begin{aligned} \sqrt{ \int _\varOmega c\phi ^2}\le \sqrt{ \int _\varOmega | \nabla \phi |^2}+\sqrt{ \int _\varOmega \frac{b^2}{4}\phi ^2},\quad \phi \in C_c^\infty (\varOmega ). \end{aligned}$$

We consider here three-dimensional water flows governed by the geophysical water wave equations exhibiting full Coriolis and centripetal terms. More precisely, assuming a constant vorticity vector, we derive a family of explicit solutions, in Eulerian coordinates, to the above-mentioned equations and their boundary conditions. These solutions are the only ones under the assumption of constant vorticity

We give necessary and sufficient conditions for the existence of entire solutions bounded or large of the equation \({\mathcal {L}}u - p\psi (u) =0\), where \({\mathcal {L}}\) is either the Laplace operator on \({\mathbb {R}} ^d\), \(d\ge 3\) or the Laplace–Beltrami operator on the harmonic NA group and p is a function whose oscillation tends to zero at infinity at a specified rate. The results apply

This paper studies the boundary behaviour of \(\lambda \)-polyharmonic functions for the simple random walk operator on a regular tree, where \(\lambda \) is complex and \(|\lambda |> \rho \), the \(\ell ^2\)-spectral radius of the random walk. In particular, subject to normalisation by spherical, resp. polyspherical functions, Dirichlet and Riquier problems at infinity are solved, and a non-tangential

In this paper, we study the supercritical problem \((P_\varepsilon )\): \( -\triangle u=K|u|^{4/(n-2)+\varepsilon }u~ \text{ in } \Omega , ~ u=0 \text{ on } \partial \Omega ,\) where \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^n\), \(n=3,4\), \(\varepsilon \) is a positive real parameter and K is a \(C^4\) positive function on \({\overline{\Omega }}\). Following the ideas of Bahri et al

We completely classify Levi flat CR structures (that is, CR structures with vanishing Levi form) on three-dimensional real Lie algebras, in terms of their algebraic and almost contact properties.